- 1. Mathematics in Middle and Upper Primary Fall EDUC8505 08 Rationale Mathematics is a discipline which has evolved from the human need to measure and communicate about time, quantity and space (Moursund, 2002). It is inherently abstract, applicable over a wide field and uses symbols to represent mathematical concepts. Traditional theoretical frameworks associated with children’s mathematical thinking include empiricism, where knowledge is external and acquired through the senses, (neo)nativism/rationalism which emphasises the in-born capabilities of the child to reason, and interactionalism, which recognises interacting roles of nature and experience, and considers the child as active in knowledge construction (Lester, 2007). A central part of each of these frameworks is experiences, which allow children to internalise or express knowledge. Experiences provide opportunities to learn, which are considered “the single most important predictor of student achievement” (National Research Council, 2001, p334; cited in Lester, 2007), and allow children to acquire physical, socio-conventional and logico-mathematical knowledge (Piaget, 1967, cited in Kamii, 2004). They are instrumental in supporting student affect, which plays a crucial role in mathematics teaching and learning (Hart&Walker, 1993, cited in Baroody, 1998). Experiences should illustrate a wide variety of examples relating to the key concept in different contexts, to facilitate students forming multiple representations and connections, and building conceptual understanding, rather than simply applying procedural knowledge. Brownell (cited in Lester, 2007) refers to conceptual understanding as mental connections among mathematical facts, procedures and ideas. Vergnaud (1983, cited in Lester, 2007) introduced the concept of the multiplicative conceptual field, a complex system of interrelated concepts, student ideas (competencies and misconceptions), procedures, problems, representations, objects, properties and relationships that cannot be studied in isolation, including multiplication, division, fractions, ratios, simple and multiple proportions, rational numbers, dimensional analysis and vector spaces. The programme provides varied activities representing the five sub-constructs of fractions (Kieren, 1980, cited in Way Sharon McCleary 5
- 2. Mathematics in Middle and Upper Primary EDUC8505 & Bobis, 2011) part-whole, measure, quotient, operator, ratio), and encourages connections between different rational numbers (e.g. Lesson 6: Fractions as quotients, incorporating relational concepts and using numbers with common factors, which support richer interconnections (Empson, 2005)). Planned experiences should also aim to expose misconceptions, prevent the formation of new ones (Bottle, 2005), and cater to students of different ability levels by using open-ended activities. The programme achieves this by selecting activities that target common misconceptions and require students to disprove them with concrete materials (e.g. Lesson 3: Show Me A Half). Concrete materials are central to assisting students in the concrete operations phase of development understand mathematical concepts (Kamii, 2004). However their use does not automatically result in mathematical learning, as students can focus on unintended aspects and fail to abstract the intended concept (Gray, 1999). Consequently, opportunities for exploring the manipulative before using it are given (Lesson 4: Exploring Pattern Blocks), and use is closely aligned with conceptual understanding and linked to symbolic conventions in order to promote purposeful connections in students’ minds. Several frameworks characterise mathematical learning as progressing from physical/concrete interaction, to generalising abstract ideas/concepts and representing them symbolically (Cowan, 2006; Lester, 2007; Baroody, 1998). Visual imagery constructed from concrete experiences is central to this progression, and its role in assisting learning has been addressed by several researchers, including Presmeg, Goldin and Thomas. Consequently, several experiences in the programme use concrete materials to encourage clear visual images that may assist children in thinking mathematically (e.g. Lesson 1: water in glasses, Lessons 4&5: Pattern Blocks). Correct mathematical language and writing conventions are also crucial to this process, since effective communication is pivotal in clarifying inconsistencies between the child’s inner understandings and correct conceptual understanding, and in allowing opportunities for exchanges between peers, expanding strategy knowledge through social learning (Vygotsky, 1978). The teacher’s role is to provide clear links Sharon McCleary 2
- 3. Mathematics in Middle and Upper Primary EDUC8505 between concepts and conventional language/symbols, enabling semiotic meaning making without stifling inherent thought processes. Opportunities to build fluency are provided in each lesson of the programme through reading, writing, talking and listening. Another feature linked to developing conceptual understanding is allowing students to actively expend effort in making sense of important mathematical ideas. Festinger’s (1957) theory of cognitive dissonance describes perplexity as a central impetus for cognitive growth, and Hatano (1988) identifies cognitive incongruity as the critical trigger for developing reasoning skills that display conceptual understanding (Lester, 2007). This is consistent with constructivist ideas of presenting problems near the boundary of the student’s Zone of Proximal Development (Vygotsky, 1978), allowing sufficient challenge to promote thinking and application of conceptual knowledge, while supporting opportunities for success and maintaining positive affect: “Acquired knowledge is most useful to a learner when it is discovered through their own cognitive efforts, related to and used in reference to what one has known before” (Bruner, cited in Cowan, 2006, pg 26). The programme incorporates problem solving allowing different solution methods: Lessons 6 & 12 provide additive and multiplicative thinking arising from invented strategies for division questions, allowing students opportunities to ‘struggle’ with relevant mathematical concepts in authentic scenarios. The teacher’s role encompasses providing children with engaging, challenging and enjoyable experiences which emphasise conceptual understanding and promote a positive attitude towards mathematics. Implicit in this is creating a classroom environment which allows opportunities for discussion, assists students in becoming fluent with conventional mathematical language/symbols and is accepting of invented strategies and solution methods. This facilitates students’ forming connections between multiple representations and abstracting meaning from experiences to progress and apply their mathematical thinking. (810 words) Sharon McCleary 3
- 4. Mathematics in Middle and Upper Primary EDUC8505 References Baroody, A. & Coslick, R. (1998). Fostering Children’s Mathematical Power, An Investigative Approach to K-8 Mathematics Instruction. Lawrence Erlbaum Associates, London. Bottle, G. (2005). Teaching Mathematics in the Primary School. Continuum, London. Burns, M. (2001). Lessons for Introducing Fractions. Math Solution Publications. California. Cathcart, W., Pothier, Y., Vance, J. & Bezuk, N. (2011). Learning Mathematics in Elementary and Middle Schools, A Learner-Centered Approach. 5th Edition. Pearson Education, Boston. Clarke, D. & Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. DOI: 10.1007/s10649-009-9198-9. Curriculum Council (Ed.). (1998). Curriculum Framework, Kindergarten to Year 12 Education in Western Australia (Mathematics Learning Area Statement). Curriculum Council of Western Australia. Perth. WA. Retrieved from http://www.curriculum.wa.edu.au Curriculum Council. (2005). Outcomes and Standards Framework and Syllabus Documents, Progress Maps and Curriculum Guide. Curriculum Council of Western Australia. Perth. WA. Retrieved from http://www.curriculum.wa.edu.au Confrey, J. & Carrejo, D. (2005). Chapter 4: Ratio and Fraction: The Difference Between Epistemological Complementarity and Conflict. Journal for Research in Mathematics Education. Sharon McCleary 5
- 5. Mathematics in Middle and Upper Primary EDUC8505 Copeland, R. (1970). How Children Learn Mathematics, Teaching Implications of Piaget’s Research, The Macmillan Company, London. Cowan, P. (2006). Teaching Mathematics, A Handbook for Primary & Secondary School Teachers, Routledge, New York. Department of Education Victoria. (2011). Fractions and Decimals Online Interview Classroom Activities. Retrieved from http://www.education.vic.gov.au Department of Education and Training Western Australia. (2007). Middle Childhood: Mathematics/ Number Scope and Sequence. Retrieved from: http://www.curriculum.wa.edu.au/internet/Years_K10/Curriculum_Resources Devlin, K. (2006). Mathematical Association of America, How do we learn math?. Retrieved from: www.maa.org/devlin/devlin_03_06.html Dienes, Z.P. (1973). Mathematics through the senses, games, dance and art, NFER Publishing Company, Ltd, New York. Downton, A., Knight, R., Clarke, D. & Lewis, G. (2006). Mathematics Assessment for Learning: Rich Tasks & Work Samples. Mathematics Teaching and Learning Centre. Melbourne. Australia. Empson, S.B., Junk, D., Dominguez, H., & Turner, E. (2005). Fractions as the co- ordination of multiplicatively related quantities: a cross-sectional study of children’s thinking. Educational Studies in Mathematics 63, pg1-28. Flewelling, G., Lind, J. & Sauer, R. (2010). Rich Learning Tasks in Number. The Australian Association of Mathematics Teachers. South Australia. Fraser, C. (2004). The development of the common fraction concept in Grade 3 learners. Pythagoras 59. pg26-33. Gray, E., Pitta, D. & Tall, D. (1999). Objects, Actions and Images: A Perspective on Early Number Development. Mathematics Education Research Centre, Coventry, UK. Sharon McCleary 5
- 6. Mathematics in Middle and Upper Primary EDUC8505 Halberda, J. & Feigenson, L. (2008). Developmental Change in the Acuity of the “Number Sense”: The Approximate Number System in 3-, 4-, 5-, and 6-Year- Olds and Adult. Developmental Psychology, Vol. 44, No. 5, pg 1457-1465. Kamii, C. (1984). Autonomy as the aim of childhood education: A Piagetian Approach, Galesburg, IL. Kamii, C. (2004). Young Children Continue to Reinvent Arithmetic – 2nd Grade- Implications of Piaget’s Theory, 2nd Edition, Teachers College Press, London. Lappan, G., Fey, J., Fitzgerald, W., Friel, S & Phillips, E. (2002). Bits and Pieces I Understanding Rational Numbers. Prentice Hall, Illinois. Lester, F. (Ed.) (2007). Second Handbook of Research on Mathematics Teaching and Learning. National Council of Teachers of Mathematics, USA. McClure, L. (2005). Raising the Profile, Whole School Maths Activities for Primary Pupils. The Mathematical Association. Leicester. McIntosh, A., Reys, B., Reys, R. & Hope, J. (1997). NumberSENSE: Simple Effect Number Sense Experiences, Dale Seymour Publications, USA. Moseley B. (2005). Students’ Early Mathematical Representation Knowledge: The Effects of Emphasizing Single or Multiple Perspectives of the Rational Number Domain in Problem Solving. Moss, J. & Case, R. (1999). Developing Children’s Understanding of the Rational Numbers: A New Model and an Experimental Curriculum. Journal for Research in Mathematics Education. Vol. 30. No. 2. Pp122-47. Moursand, D., (2006), Mathematics, Retrieved from: http://darkwing.uoregon.edu/~moursund/math/mathematics.htm Muir, T. (2008). Principles of Practice and Teacher Actions: Influences on Effective Teaching of Numeracy. Mathematics Education Research Journal. Vol. 20, No. 3, pg 78-101. Nunes, T. & Bryant, P. (1996). Children Doing Mathematics. Blackwell Publishers, Massachusetts, USA. Presmeg, N. (n.d.). Research on Visualisation in Learning and Teaching Sharon McCleary 6
- 7. Mathematics in Middle and Upper Primary EDUC8505 Mathematics, Illinois State University. Radford, L., Schubring, G. & Seeger, F. (2011). Signifying and meaning-making in mathematical thinking, teaching an learning. DOI: 10.1007/s10649-011-9322-5. Reys, R. & Yang, D.C. (1998). Relationship Between Computational Performance and Number Sense Among Sixth- and Eighth-Grade Students in Taiwan. Journal for Research in Mathematics Education. Vol. 29, No. 2, pg 225-237. Schneider, M., Grabner, R. & Paetsch, J. (2009). Mental Number Line, Number Line Estimation, and Mathematical Achievement: Their Interrelations in Grads 5 and 6. Journal of Educational Psychology. Vol. 101, No. 2. pgs 359- 372. Siegler, R., Thompson, C. & Schneider, M. (2011). An Integrated theory of whole number and fractions development. Cognitive Psychology. Vol. 62. pp273- 296. Smith, C., Solomon, G. & Carey, S. (2005). Never getting to zero: elementary school students’ understanding of the infinite divisibility of number and matter. Cognitive Psychology. Vol.51. pp101-140. Stenmark, J. & Bush, W. (2001). Mathematics Assessment, A Practical Handbook. National Council of Teachers of Mathematics. VA. Sullivan, P. & Lilburn, P. (2004). Open-ended Maths Activities, Using ‘good’ questions to enhance learning in Mathematics. 2nd Edition, Oxford University Press, Oxford. The Australian Curriculum-Mathematics, Version 1.1, (2010). Australian Curriculum, Assessment and Reporting Authority [ACARA], Retrieved from: http://www.australiancurriculum.edu.au Vygotsky, L., (1978). Mind in Society, Harvard University Press, Cambridge, MA. Way, J. & Bobis, J. (2011). Fractions, Teaching for Understanding. The Australian Association of Mathematics Teachers Inc. South Australia. Wilkerson-Jerde, M. & Wilensky, U. (2011). How do mathematicians learn math?: resources and acts for constructing and understanding mathematicians, doi: 10.1007/s10649-011-9306-5. Sharon McCleary 7
- 8. Mathematics in Middle and Upper Primary EDUC8505 Willis, S., Devlin, W., Jacob, L., Powell, B., Tomazos, D. & Treacy, K. (2004), First Steps in Mathematics: Number (Book 1). Rigby. Australia. Sharon McCleary 8
- 9. Mathematics in Middle and Upper Primary EDUC8505 References Baroody, A. & Coslick, R. (1998). Fostering Children’s Mathematical Power, An Investigative Approach to K-8 Mathematics Instruction. Lawrence Erlbaum Associates, London. Bottle, G. (2005). Teaching Mathematics in the Primary School. Continuum, London. Burns, M. (2001). Lessons for Introducing Fractions. Math Solution Publications. California. Cathcart, W., Pothier, Y., Vance, J. & Bezuk, N. (2011). Learning Mathematics in Elementary and Middle Schools, A Learner-Centered Approach. 5th Edition. Pearson Education, Boston. Clarke, D. & Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. DOI: 10.1007/s10649-009-9198-9. Curriculum Council (Ed.). (1998). Curriculum Framework, Kindergarten to Year 12 Education in Western Australia (Mathematics Learning Area Statement). Curriculum Council of Western Australia. Perth. WA. Retrieved from http://www.curriculum.wa.edu.au Curriculum Council. (2005). Outcomes and Standards Framework and Syllabus Documents, Progress Maps and Curriculum Guide. Curriculum Council of Western Australia. Perth. WA. Retrieved from http://www.curriculum.wa.edu.au Confrey, J. & Carrejo, D. (2005). Chapter 4: Ratio and Fraction: The Difference Between Epistemological Complementarity and Conflict. Journal for Research in Mathematics Education. Sharon McCleary 5
- 10. Mathematics in Middle and Upper Primary EDUC8505 Copeland, R. (1970). How Children Learn Mathematics, Teaching Implications of Piaget’s Research, The Macmillan Company, London. Cowan, P. (2006). Teaching Mathematics, A Handbook for Primary & Secondary School Teachers, Routledge, New York. Department of Education Victoria. (2011). Fractions and Decimals Online Interview Classroom Activities. Retrieved from http://www.education.vic.gov.au Department of Education and Training Western Australia. (2007). Middle Childhood: Mathematics/ Number Scope and Sequence. Retrieved from: http://www.curriculum.wa.edu.au/internet/Years_K10/Curriculum_Resources Devlin, K. (2006). Mathematical Association of America, How do we learn math?. Retrieved from: www.maa.org/devlin/devlin_03_06.html Dienes, Z.P. (1973). Mathematics through the senses, games, dance and art, NFER Publishing Company, Ltd, New York. Downton, A., Knight, R., Clarke, D. & Lewis, G. (2006). Mathematics Assessment for Learning: Rich Tasks & Work Samples. Mathematics Teaching and Learning Centre. Melbourne. Australia. Empson, S.B., Junk, D., Dominguez, H., & Turner, E. (2005). Fractions as the co- ordination of multiplicatively related quantities: a cross-sectional study of children’s thinking. Educational Studies in Mathematics 63, pg1-28. Flewelling, G., Lind, J. & Sauer, R. (2010). Rich Learning Tasks in Number. The Australian Association of Mathematics Teachers. South Australia. Fraser, C. (2004). The development of the common fraction concept in Grade 3 learners. Pythagoras 59. pg26-33. Gray, E., Pitta, D. & Tall, D. (1999). Objects, Actions and Images: A Perspective on Early Number Development. Mathematics Education Research Centre, Coventry, UK. Sharon McCleary 10
- 11. Mathematics in Middle and Upper Primary EDUC8505 Halberda, J. & Feigenson, L. (2008). Developmental Change in the Acuity of the “Number Sense”: The Approximate Number System in 3-, 4-, 5-, and 6-Year- Olds and Adult. Developmental Psychology, Vol. 44, No. 5, pg 1457-1465. Kamii, C. (1984). Autonomy as the aim of childhood education: A Piagetian Approach, Galesburg, IL. Kamii, C. (2004). Young Children Continue to Reinvent Arithmetic – 2nd Grade- Implications of Piaget’s Theory, 2nd Edition, Teachers College Press, London. Lappan, G., Fey, J., Fitzgerald, W., Friel, S & Phillips, E. (2002). Bits and Pieces I Understanding Rational Numbers. Prentice Hall, Illinois. Lester, F. (Ed.) (2007). Second Handbook of Research on Mathematics Teaching and Learning. National Council of Teachers of Mathematics, USA. McClure, L. (2005). Raising the Profile, Whole School Maths Activities for Primary Pupils. The Mathematical Association. Leicester. McIntosh, A., Reys, B., Reys, R. & Hope, J. (1997). NumberSENSE: Simple Effect Number Sense Experiences, Dale Seymour Publications, USA. Moseley B. (2005). Students’ Early Mathematical Representation Knowledge: The Effects of Emphasizing Single or Multiple Perspectives of the Rational Number Domain in Problem Solving. Moss, J. & Case, R. (1999). Developing Children’s Understanding of the Rational Numbers: A New Model and an Experimental Curriculum. Journal for Research in Mathematics Education. Vol. 30. No. 2. Pp122-47. Moursand, D., (2006), Mathematics, Retrieved from: http://darkwing.uoregon.edu/~moursund/math/mathematics.htm Muir, T. (2008). Principles of Practice and Teacher Actions: Influences on Effective Teaching of Numeracy. Mathematics Education Research Journal. Vol. 20, No. 3, pg 78-101. Nunes, T. & Bryant, P. (1996). Children Doing Mathematics. Blackwell Publishers, Massachusetts, USA. Presmeg, N. (n.d.). Research on Visualisation in Learning and Teaching Sharon McCleary 11
- 12. Mathematics in Middle and Upper Primary EDUC8505 Mathematics, Illinois State University. Radford, L., Schubring, G. & Seeger, F. (2011). Signifying and meaning-making in mathematical thinking, teaching an learning. DOI: 10.1007/s10649-011-9322-5. Reys, R. & Yang, D.C. (1998). Relationship Between Computational Performance and Number Sense Among Sixth- and Eighth-Grade Students in Taiwan. Journal for Research in Mathematics Education. Vol. 29, No. 2, pg 225-237. Schneider, M., Grabner, R. & Paetsch, J. (2009). Mental Number Line, Number Line Estimation, and Mathematical Achievement: Their Interrelations in Grads 5 and 6. Journal of Educational Psychology. Vol. 101, No. 2. pgs 359- 372. Siegler, R., Thompson, C. & Schneider, M. (2011). An Integrated theory of whole number and fractions development. Cognitive Psychology. Vol. 62. pp273- 296. Smith, C., Solomon, G. & Carey, S. (2005). Never getting to zero: elementary school students’ understanding of the infinite divisibility of number and matter. Cognitive Psychology. Vol.51. pp101-140. Stenmark, J. & Bush, W. (2001). Mathematics Assessment, A Practical Handbook. National Council of Teachers of Mathematics. VA. Sullivan, P. & Lilburn, P. (2004). Open-ended Maths Activities, Using ‘good’ questions to enhance learning in Mathematics. 2nd Edition, Oxford University Press, Oxford. The Australian Curriculum-Mathematics, Version 1.1, (2010). Australian Curriculum, Assessment and Reporting Authority [ACARA], Retrieved from: http://www.australiancurriculum.edu.au Vygotsky, L., (1978). Mind in Society, Harvard University Press, Cambridge, MA. Way, J. & Bobis, J. (2011). Fractions, Teaching for Understanding. The Australian Association of Mathematics Teachers Inc. South Australia. Wilkerson-Jerde, M. & Wilensky, U. (2011). How do mathematicians learn math?: resources and acts for constructing and understanding mathematicians, doi: 10.1007/s10649-011-9306-5. Sharon McCleary 12
- 13. Mathematics in Middle and Upper Primary EDUC8505 Willis, S., Devlin, W., Jacob, L., Powell, B., Tomazos, D. & Treacy, K. (2004), First Steps in Mathematics: Number (Book 1). Rigby. Australia. Sharon McCleary 13